Problem 32 - College Algebra

Question

express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. 4 and 15

Step-by-Step Solution

Answer

The distance between 4 and 15 is 11.

Step 1: Identify the two numbers

The two numbers given are 4 and 15.

Step 2: Calculate the difference

Subtract one number from the other. It doesn't matter which order this is done in because the absolute value will remove any negative sign. 15 - 4 gives 11.

Step 3: Take the absolute value

The absolute value of 11 is 11. The distance between the numbers 4 and 15 is 11.

Key Concepts

Distance Between Numbers

When we talk about the distance between two numbers, we're really discussing how far apart these numbers are on a number line.
Imagine the number line is like a ruler that stretches from left to right with numbers placed in a sequence.
The concept of distance here is quite simple: It's the total space or gap between the two given numbers. Unlike physical distance, where we account for left or right, on a number line, distance is always positive.

• Think of it like this: If you move from 4 to 15 on the number line, you are covering the space of 11 units.
• Whether you measure from left to right or right to left, the distance remains constant.

This is why we use absolute value in distance calculations: it ensures the number is positive, giving us the correct measure of space between numbers.

Evaluate Absolute Value

Evaluating the absolute value of a number is a simple process that helps in determining the magnitude without considering any negative signs.
In mathematical terms, the absolute value of a number is how far that number is from zero on the number line.
To "evaluate" means to find the value of this expression. Taking the absolute value is straightforward:

• If the original number is positive or zero, the absolute value is the same as the original number.
• If the original number is negative, the absolute value is its positive counterpart.

For example, if you subtract 4 from 15, you get 11. The absolute value of 11 is still 11 because it’s already positive.
This means the distance—as evaluated through absolute value—gives you a clear picture without worrying about direction. Absolute value serves as an essential tool to highlight only the size or scale of the distance.

Number Line

A number line is a straight line that represents numbers in equal intervals. It acts as a visual aid that can make understanding numbers and their relationships much easier.
By using a number line, you can easily see where numbers are located relative to one another and measure the distance between them.

• Each point on the line corresponds to a single number.
• Moving to the right indicates increasing value, while moving to the left indicates decreasing value.

This visual representation can simplify concepts of addition, subtraction, and absolute value. When considering absolute value on a number line, you focus only on how many units a number is away from zero—always counting in a non-negative direction.

In our example, you could easily visualize 4 and 15 on the number line, measure the units between them, and confirm that they are 11 units apart, showcasing the distance in an intuitive way. Using a number line for such exercises not only makes the task more visual but also reinforces the core understanding of numerical distance.